A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts
Abstract
:1. Introduction
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- rotation with the angle φ1 of the tulip with respect to the axis X1, φ1 = 0 … n1π;
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- rotation with the angle φ2 of the mid shaft with respect to the axis X2, φ2 = 0 … n1π;
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- rotation with the angle φ3 of the bowl with respect to the axis X3, φ3 = 0 … n1π;
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- relative rotation of the longitudinal axe of the midshaft (given by the direction of the axis X2) with respect to the longitudinal direction of the tulip (given by the direction of the axis X1), with β1 (spatial angle between axis X1 and X2) with respect to the axis Z1, β1 being the angle between longitudinal direction of the tulip and the longitudinal direction of the midshaft, β1 = 0° … 15°;
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- relative rotation of the longitudinal axis of the bowl (given by the direction of the axis X3) with respect to the longitudinal direction of the midshaft (given by the direction of the axis X2), with β2 (spatial angle between axis X2 and X3) with respect to the axe Y2, β2 being the angle between the longitudinal direction of the midshaft and the longitudinal direction of the bowl, β2 = 0° … 47°.
2. Computation of the Mass Moments and Geometric Moments of Driveshaft Inertia
3. The Physical Model of the Driveshaft in Torsion
- the tulip in torsional rigid body movement reduced to the torsional longitudinal axis of the midshaft, having a global torsional stiffness , a global torsional damping coefficient , an axial geometric moment of inertia of the cross section for the global tulip reduced to the longitudinal axis of the midshaft in the centroid of the cross section of tripode fixed on the midshaft (see Equation (1)), an axial mass moment of inertia of the cross section for the global tulip reduced to the longitudinal axis of the midshaft in the centroid of the cross section of tripode fixed on the midshaft (see Equation (5)), where and are given by the equations:
- The joint tulip–tripod in torsion realizes the link between the tulip and the midshaft through the torsional stiffness and the damping torsional coefficient ;
- The uniform midshaft (see Figure 2, Figure 4, and Figure 5) in torsion having, at x = 0, a tripod (see Figure 4) fixed on the midshaft with the axial mass moment of inertia of the cross section (midshaft axis included on the thickness of the tripod) and the geometric axial moment of inertia of the cross section of the tripod (midshaft axis included on the thickness of the tripod), and at , an inner race (see Figure 5) fixed on the midshaft with the axial mass moment of inertia of the cross section (midshaft axis included on the thickness of the inner race) and the geometric axial moment of inertia of the cross section (midshaft axis included on the thickness of the inner race), given by the equations:
- The joint bowl–balls–inner race in torsion realizes the link between the bowl and the midshaft through the torsional stiffness ktBIr and the damping torsional coefficient ctBIr;
- The bowl in torsional rigid body movement is reduced to the torsional longitudinal axis of the midshaft, having a global torsional stiffness , a global torsional damping coefficient , an axial geometric moment of inertia of the cross section reduced to the longitudinal axe of the midshaft JX2GB (see Equation (6), an axial mass moment of inertia of the cross section reduced to the longitudinal axe of the midshaft (see Equation (10)), where and are given by the equations:
4. The Equations of Forced Torsional Vibrations of the Automotive Driveshaft
5. The Mathematical Procedure Solution
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- primary resonances for excitation frequencies [25] (p. 196),
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- super harmonic resonances for excitation frequencies , k1, k2, positive integers [25] (p. 211),
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- subharmonic resonances for excitation frequencies k1, k2, positive integers [25] (p. 214),
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- principal parametric resonances for excitation frequencies [25] (p. 425),
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- combination resonances for excitation frequencies [25] (pp. 202, 430),
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- simultaneous resonances for excitation frequencies , with k positive integer [25] (p. 188),
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- internal resonances for with k1, k2, positive integers [25] (p. 381), being the excitation frequency. As can be seen, this model for the torsional forced vibrations of driveshaft offer a huge possibility of investigation.
6. Case Study Analysis of Principal Parametric Resonance of the Global Tulip
7. Results and Discussions
8. Conclusions
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- nonuniformity in the geometric and kinematic isometry of the driveshaft;
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- nonuniformity in the geometric and mass moments of inertia of the cross section for the tulip, tripod, inner race, and bowl;
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- the stiffness and the damping link of the joints of the driveshaft tulip–tripod–midshaft and midshaft–inner race–balls–bowl;
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- harmonic excitation of the driveshaft due to the car engine;
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- impulsive excitation of the driveshaft due to road excitation.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Mathematical Expressions of the Coefficients Used in the Solution Algorithm
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LT [m] | LAT [m] | LMs [m] | RTTR [m] | dAT [m] | dCT [m] | ST [m2] | 0.5(J1T + J2T) [m4] | |
---|---|---|---|---|---|---|---|---|
0.095 | 0.065 | 0.470 | 0.035 | 0.027 | 0.049 | 0.019 | 9.1531 × 10−7 | 0.15 |
[kg/m3] | G [GPa] | Torsional Rigidity [Nm/rad] | Damping Ratio | Engine Torque [Nm] |
---|---|---|---|---|
7850 | 77.3 | 1.11 × 104 | 0.0016–0.0318 | 580 |
Order | 1 | 2 | 3 |
---|---|---|---|
[Hz] | 3338.31 | 6676.63 | 10,014.94 |
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Bugaru, M.; Vasile, A. A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts. Computation 2022, 10, 10. https://0-doi-org.brum.beds.ac.uk/10.3390/computation10010010
Bugaru M, Vasile A. A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts. Computation. 2022; 10(1):10. https://0-doi-org.brum.beds.ac.uk/10.3390/computation10010010
Chicago/Turabian StyleBugaru, Mihai, and Andrei Vasile. 2022. "A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts" Computation 10, no. 1: 10. https://0-doi-org.brum.beds.ac.uk/10.3390/computation10010010